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Properties

Label 2-980-140.139-c1-0-68
Degree $2$
Conductor $980$
Sign $0.997 + 0.0747i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.940 − 1.05i)2-s + 2.59i·3-s + (−0.232 − 1.98i)4-s + (1.75 − 1.38i)5-s + (2.74 + 2.44i)6-s + (−2.31 − 1.62i)8-s − 3.74·9-s + (0.194 − 3.15i)10-s + 3.60i·11-s + (5.15 − 0.603i)12-s + 0.818·13-s + (3.58 + 4.56i)15-s + (−3.89 + 0.924i)16-s + 7.39·17-s + (−3.51 + 3.95i)18-s + 3.30·19-s + ⋯
L(s)  = 1  + (0.664 − 0.747i)2-s + 1.49i·3-s + (−0.116 − 0.993i)4-s + (0.786 − 0.617i)5-s + (1.11 + 0.996i)6-s + (−0.819 − 0.573i)8-s − 1.24·9-s + (0.0614 − 0.998i)10-s + 1.08i·11-s + (1.48 − 0.174i)12-s + 0.226·13-s + (0.925 + 1.17i)15-s + (−0.972 + 0.231i)16-s + 1.79·17-s + (−0.828 + 0.931i)18-s + 0.758·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.997 + 0.0747i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.997 + 0.0747i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62425 - 0.0982803i\)
\(L(\frac12)\) \(\approx\) \(2.62425 - 0.0982803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.940 + 1.05i)T \)
5 \( 1 + (-1.75 + 1.38i)T \)
7 \( 1 \)
good3 \( 1 - 2.59iT - 3T^{2} \)
11 \( 1 - 3.60iT - 11T^{2} \)
13 \( 1 - 0.818T + 13T^{2} \)
17 \( 1 - 7.39T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 2.04T + 29T^{2} \)
31 \( 1 - 1.91T + 31T^{2} \)
37 \( 1 + 7.16iT - 37T^{2} \)
41 \( 1 - 2.65iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 - 1.33iT - 47T^{2} \)
53 \( 1 + 1.81iT - 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 - 9.77iT - 61T^{2} \)
67 \( 1 + 9.26T + 67T^{2} \)
71 \( 1 + 1.38iT - 71T^{2} \)
73 \( 1 + 7.39T + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 7.32T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.10440823898302870711871214563, −9.458110653087492861102501094168, −8.951497283055260211250176360185, −7.47170093429546982676624545776, −5.98318772438101639021198766600, −5.31224178138279762496296727166, −4.69279945956591710139105243100, −3.81265112369859108015633681889, −2.81361598659535311402991825049, −1.37092782439064304519807500887, 1.25340829852258945469668908315, 2.74948328321752375663026834564, 3.43409510796204093181004682981, 5.27857118919037146167466174467, 5.90565865307289788616223075682, 6.53179213692723993206884199338, 7.37768966925727412022688124239, 7.936340821323149431455081394047, 8.838463287592884896779543885544, 9.929533415637876663932292621185

Graph of the $Z$-function along the critical line